Intro to Accelerator Physics

1. Intro to Accelerator Physics Eric Prebys Fermi National Accelerator Laboratory NIU Phys 790 Guest Lecture E. Prebys, NIU Phy 790 Guest Lecture

2. Relativity and Units Some Handy Relationships These units make these relationships really easy to calculate E. Prebys, NIU Phy 790 Guest Lecture • Basic Relativity • Units • For the most part, we will use SI units, except • Energy: eV (keV, MeV, etc) [1 eV = 1.6x10-19 J] • Mass: eV/c2 [proton = 1.67x10-27 kg = 938 MeV/c2] • Momentum: eV/c [proton @ β=.9 = 1.94 GeV/c]

3. Man-made Particle Acceleration The simplest accelerators accelerate charged particles through a static electricfield. Example: vacuum tubes (or CRT TV’s) Cathode Anode • Limited by magnitude of static field: • - TV Picture tube ~keV- X-ray tube ~10’s of keV- Van de Graaf ~MeV’s • Solutions: • Alternate fields to keep particles in accelerating fields -> RF acceleration • Bend particles so they see the same accelerating field over and over -> cyclotrons, synchrotrons FNAL Cockroft-Walton = 750 kV

4. The Cyclotron (1930’s) top view side view • A charged particle (v<<c) in a uniform magnetic field will follow a circular path of radius “Cyclotron Frequency” For a proton: Accelerating “DEES” E. Prebys, NIU Phy 790 Guest Lecture

5. Round and Round We Go: the First Cyclotrons • ~1930 (Berkeley) • Lawrence and Livingston • K=80KeV • 1935 - 60” Cyclotron • Lawrence, et al. (LBL) • ~19 MeV (D2) • Prototype for many E. Prebys, NIU Phy 790 Guest Lecture

6. Beam “rigidity” top view side view • The relativistic form of Newton’s Laws for a particle in an electromagnetic field is: • A particle of unit charge in a uniform magnetic field will move in a circle of radius constant for fixed energy units of eV in our usual convention T-m2/s=V Beam “rigidity” = constant at a given momentum (even when B=0!) Remember forever! E. Prebys, NIU Phy 790 Guest Lecture

7. Application: Thin Lens Approximation and Magnetic “kick” E. Prebys, NIU Phy 790 Guest Lecture If the path length through a transverse magnetic field is short compared to the bend radius of the particle, then we can think ofthe particle receiving a transverse “kick”and it will be bent through small angle In this “thin lens approximation”, a dipole is the equivalent of a prism in classical optics.

8. Synchrotrons ”Synchrotron”* *Edward McMillan, 1945 E. Prebys, NIU Phy 790 Guest Lecture • Cyclotrons have a fixed magnetic field • As the energy grows, so does the radius • Since the minimum radius is limited by the strength of the magnetic fields we can make, cyclotron get impractically large pretty quickly. • On the other hand, we just showed the trajectory of a particle will always depend on • If, as the particles accelerate, we scale all magnetic fields with the particle momentum, then the trajectories will be independent of energy!

9. Weak Focusing E. Prebys, NIU Phy 790 Guest Lecture • Cyclotrons relied on the fact that magnetic fields between two pole faces are never perfectly uniform. • This prevents the particles from spiraling out of the pole gap. • In early synchrotrons, radial field profiles were optimized to take advantage of this effect, but in any weak focused beams, the beam size grows with energy. • The highest energy weak focusing accelerator was the Berkeley Bevatron, which had a kinetic energy of 6.2 GeV • High enough to make antiprotons(and win a Nobel Prize) • It had an aperture 12”x48”!

10. Strong Focusing E. Prebys, NIU Phy 790 Guest Lecture • Strong focusing utilizes alternating magnetic gradients to precisely control the focusing of a beam of particles • The principle was first developed in 1949 by Nicholas Christofilos, a Greek-American engineer, who was working for an elevator company in Athens at the time. • Rather than publish the idea, he applied for a patent, and it went largely ignored. • The idea was independently invented in 1952 by Courant, Livingston and Snyder, who later acknowledged the priority of Christophilos’ work. • Although the technique was originally formulated in terms of magnetic gradients, it’s much easier to understand in terms of the separate functions of dipole and quadrupole magnets.

11. Combined Function vs. Separated Function Strong focusing was originally implemented by building magnets with non parallel pole faces to introduce a linear magnetic gradient = + dipole quadrupole CERN PS (1959, 29 GeV) Later synchrotrons were built with physically separate dipole and quadrupole magnets. The first “separated function” synchrotron was the Fermilab Main Ring (1972, 400 GeV) = + dipole quadrupole Fermilab Strong focusing is also much easier to teach using separated functions, so we will… E. Prebys, NIU Phy 790 Guest Lecture

12. Quadrupole Magnets as Lenses* • A positive particle coming out of the page off center in the horizontal plane will experience a restoring kick focal length *or quadrupole term in a gradient magnet E. Prebys, NIU Phy 790 Guest Lecture

13. What About the Other Plane? Defocusing! Luckily, if we place equal and opposite pairs of lenses, there will be a net focusing regardless of the order. pairs give net focusing in both planes -> “FODO cell” E. Prebys, NIU Phy 790 Guest Lecture

14. A (first) Word about Coordinates and Conventions For now, assume bends are always in the x-s plane Particle trajectory defined at any point in s by location in x,x’ ory,y’ “phase space” unique initial phase space point  unique trajectory E. Prebys, NIU Phy 790 Guest Lecture We will work in a right-handed coordinate system with x horizontal, y vertical, and s along the nominal trajectory.

15. Transfer Matrices Quadrupole: Drift: E. Prebys, NIU Phy 790 Guest Lecture The simplest magnetic “lattice” consists of quadrupoles and the spaces in between them (drifts). We can express each of these as a linear operation in phase space. By combining these elements, we can represent an arbitrarily complex ring or line as the product of matrices.

16. Transfer Matrix of a FODO cell f -f L L Remember: motion is usually drawn from left to right, but matrices act from right to left! E. Prebys, NIU Phy 790 Guest Lecture At the heart of every beam line or ring is the basic “FODO” cell, consisting of a focusing and a defocusing element, separated by drifts:

17. Where we’re going… • Our goal is to de-couple the problem into two parts • The “lattice”: a mathematical description of the machine itself, based only on the magnetic fields, which is identical for each identical cell • A mathematical description for the ensemble of particles circulating in the machine (“emittance”); Periodic “cell” E. Prebys, NIU Phy 790 Guest Lecture • It might seem like we would start by looking at beam lines and them move on to rings, but it turns out that there is no unique treatment of a standalone beam line • Depends implicitly in input beam parameters • Therefore, we will initially solve for stable motion in a ring. • Rings are generally periodic, made up of more or less identical cells • In addition to simplifying the design, we’ll see that periodicity is important to stability • The simplest rings are made of dipoles and FODO cells • Or “combined function magnets” which couple the two

18. Quick Review of Linear Algebra E. Prebys, NIU Phy 790 Guest Lecture • In the absence of degeneracy, an nxn matrix will have n “eigenvectors”, defined by: • Eigenvectors form an orthogonal basis • That is, any vector can be represented as a unique sum of eigenvectors • In general, there exists a unitary transformation, such that • Because both the trace and the determinant of a matrix are invariant under a unitary transformation:

19. Stability Criterion E. Prebys, NIU Phy 790 Guest Lecture We can represent an arbitrarily complex ring as a combination of individual matrices We can express an arbitrary initial state as the sum of the eigenvectors of this matrix After n turns, we have Because the individual matrices have unit determinants, the product must as well, so

20. Stability Criterion (cont’d) E. Prebys, NIU Phy 790 Guest Lecture We can therefore express the eigenvalues as However, if a has any real component, one of the solutions will grow exponentially, so the only stable values are Examining the (invariant) trace of the matrix So the general stability criterion is simply

21. Example L L f -f E. Prebys, NIU Phy 790 Guest Lecture Recall our FODO cell Our stability requirement becomes

22. Twiss Parameterization “Twiss Parameters” not Lorentz parameters!! Normalization relationship only two independent Remember this! We’ll see it again in a few pages E. Prebys, NIU Phy 790 Guest Lecture We can express the transfer matrix for one period as the sum of an identity matrix and a traceless matrix The requirement that Det(M)=1 implies We can already identify A=Tr(M)/2=cosμ. Setting the determinant of the second matrix to 1 yields the constraint We can identify B=sinμ and write Note that So we can identify it with i=sqrt(-1) and write

23. Calculating and Propagating the Lattice functions E. Prebys, NIU Phy 790 Guest Lecture If we know the transfer matrix or one period, we can explicitly calculate the lattice functions at the ends If we then know the transfer matrix from point a to point b, Mba, we can evolve the lattice functions from a to b

24. Calculating the Lattice functions (cont’d) E. Prebys, NIU Phy 790 Guest Lecture Using We can now evolve the J matrix at any point as Multiplying this mess out and gathering terms, we get

25. Examples E. Prebys, NIU Phy 790 Guest Lecture Drift of length L: Thin focusing (defocusing) lens:

26. Moving on: Equations of Motion Particle trajectory Reference trajectory s is motion along nominal trajectory Transverse acceleration = γ doesn’t change! E. Prebys, NIU Phy 790 Guest Lecture • For the moment, we will consider curvature in the horizontal (x) plane, with a reference trajectory established by the dipole fields. • General equation of motion (considering only transverse fields!) • We must solve this in the curving coordinate system • Messy but straightforward

27. Equations of Motion (cont’d) Transform independent variable from t to s s is measured along nominal trajectory, vs measured along actual trajectory Looks “kinda like” a harmonic oscillator “centripetal” term E. Prebys, NIU Phy 790 Guest Lecture We will keep only the first order terms in the magnetic field Expanding in the rotating coordinate system and keeping first order terms…

28. Comment on our Equations K(s) periodic! E. Prebys, NIU Phy 790 Guest Lecture We have our equations of motion in the form of two “Hill’s Equations” This is the most general form for a conservative, periodic, system in which deviations from equilibrium small enough that the resulting forces are approximately linear In addition to the curvature term, this can only include the linear terms in the magnetic field (ie, the “quadrupole” term) The dipole term is implicitly accounted for in the definition of the reference trajectory (local curvature ρ). Any higher order (nonlinear) terms are dealt with as perturbations. Rotated quadruple (“skew”) terms lead to coupling, which we won’t consider here.

29. General Solution E. Prebys, NIU Phy 790 Guest Lecture These are second order homogeneous differential equations, so the explicit equations of motion will be linearly related to the initial conditions by Exactly as we would expect from our initial naïve treatment of the beam line elements.

30. Piecewise Solution E. Prebys, NIU Phy 790 Guest Lecture Again, these equations are in the form For K constant, these equations are quite simple. For K>0 (focusing), it’s just a harmonic oscillator and we write In terms if initial conditions, we identifyand write

31. E. Prebys, NIU Phy 790 Guest Lecture For K<0 (defocusing), the solution becomes For K=0 (a “drift”), the solution is simply We can now express the transfer matrix of an arbitrarily complex beam line with But there’s a limit to what we can do with this

32. Closed Form Solution Phase advance over one period Super important! Remember forever! E. Prebys, NIU Phy 790 Guest Lecture Looking at our Hill’s equation If K is a constant >0, then so try a solution of the form If we plug this into the equations of motion (and do a lot of math), we find that in terms of our Twiss parameterization

33. Betatronmotion (this is the page to remember!) x s Lateral deviation in one plane The “betatron function” β(s) is effectively the local wavenumber and also defines the beam envelope. Phase advance Closely spaced strong quads -> smallβ-> small aperture, lots of wiggles Sparsely spaced weak quads -> large β-> large aperture, few wiggles E. Prebys, NIU Phy 790 Guest Lecture Generally, we find that we can describe particle motion in terms of initial conditions and a “beta function” β(s), which is only a function of location in the nominal path.

34. Behavior Over Multiple Turns Area = πA2 Particle will return to a different point on the same ellipse each time around the ring. E. Prebys, NIU Phy 790 Guest Lecture The general expressions for motion are We form the combination This is the equation of an ellipse.

35. Symmetric Treatment of FODO Cell L L 2f 2f -f Note: some textbooks have L=total length E. Prebys, NIU Phy 790 Guest Lecture If we evaluate the cell at the center of the focusing quad, it looks like Leading to the transfer Matrix

36. Lattice Functions in FODO Cell We know from our Twiss Parameterization that this can be written as From which we see that the Twiss functions at the middle of the magnets are recall Flip sign of f to get other plane E. Prebys, NIU Phy 790 Guest Lecture

37. Lattice Function in FODO Cell (cont’d) β = maxα = 0 maximum β = decreasingα >0 focusing β = minα = 0 minimum β = increasingα < 0 defocusing Motion at each point bounded by E. Prebys, NIU Phy 790 Guest Lecture As particles go through the lattice, the Twiss parameters will vary periodically:

38. Conceptual understanding of β Trajectories over multiple turns Normalized particle trajectory β(s)is also effectively the local wave number which determines the rate of phase advance Closely spaced strong quadssmall β small aperture, lots of wiggles Sparsely spaced weak quadslarge β large aperture, few wiggles E. Prebys, NIU Phy 790 Guest Lecture It’s important to remember that the betatron function represents a bounding envelope to the beam motion, not the beam motion itself

39. Betatron Tune Particle trajectory • As particles go around a ring, they will undergo a number of betatrons oscillations ν (sometimes Q) given by • This is referred to as the “tune” • We can generally think of the tune in two parts: Ideal orbit 6.7 Integer : magnet/aperture optimization Fraction: Beam Stability E. Prebys, NIU Phy 790 Guest Lecture

40. fract. part of Y tune fract. part of X tune Tune, Stability, and the Tune Plane Avoid lines in the “tune plane” “small” integers E. Prebys, NIU Phy 790 Guest Lecture If the tune is an integer, or low order rational number, then the effect of any imperfection or perturbation will tend be reinforced on subsequent orbits. When we add the effects of coupling between the planes, we find this is also true for combinations of the tunes from both planes, so in general, we want to avoid Many instabilities occur when something perturbs the tune of the beam, or part of the beam, until it falls onto a resonance, thus you will often hear effects characterized by the “tune shift” they produce.

41. Characterizing an Ensemble: Emittance If each particle is described by an ellipse with a particular amplitude, then an ensemble of particles will always remain within a bounding ellipse of a particular area: Area = ε • Either leave the π out, or include it explicitly as a “unit”. Thus • microns (CERN) and • π-mm-mr (FNAL) • Are actually the same units (just remember you’ll never have to explicity use π in the calculation) These are really the same E. Prebys, NIU Phy 790 Guest Lecture

42. Definitions of Emittance E. Prebys, NIU Phy 790 Guest Lecture • Because distributions normally have long tails, we have to adopt a convention for defining the emittance. The two most common are • Gaussian (electron machines, CERN): • 95% Emittance (FNAL): • In general, emittance can be different in the two planes, but we won’t worry about that.

43. Emittance and Beam Distributions large spatial distribution small angular distribution small spatial distribution large angular distribution E. Prebys, NIU Phy 790 Guest Lecture As we go through a lattice, the bounding emittance remains constant

44. Adiabatic Damping “Normalized emittance” =constant! E. Prebys, NIU Phy 790 Guest Lecture In our discussions up to now, we assume that all fields scale with momentum, so our lattice remains the same, but what happens to the ensemble of particles? Consider what happens to the slope of a particle as the forward momentum incrementally increases. If we evaluate the emittance at a point where α=0, we have

45. Consequences of Adiabatic Damping RMS emittance betatron function 95% emittance v/c E. Prebys, NIU Phy 790 Guest Lecture • As a beam is accelerated, the normalized emittance remains constant • Actual emittance goes down down • Which means the actual beam size goes down as well • The angular distribution at an extremum (α=0) is • We almost always use normalized emittance

46. Example: Fermilab Main Ring • First “separated function” lattice • 1 km in radius • First accelerated protons from 8 to 400 GeV in 1972 1968 E. Prebys, NIU Phy 790 Guest Lecture

47. Beam Parameters *remember this for problem set E. Prebys, NIU Phy 790 Guest Lecture The Main Ring accelerated protons from kinetic energy of 8 to 400 GeV*

48. Cell Parameters • From design report • L=29.74 m • Phase advance μ=71° • Quad Length lquad=2.13 m • Beta functions (slide 36) • Magnet focal length • Quad gradient (slide 12) E. Prebys, NIU Phy 790 Guest Lecture

49. Beam Line Calculation: MAD 98.4m (exact) vs. 99.4m (thin lens) main_ring.madx ! ! One FODO cell from the FNAL Main Ring (NAL Design Report, 1968) ! beam, particle=proton,energy=400.938272,npart=1.0E9; LQ:=1.067; LD:=29.74-2*LQ; qf: QUADRUPOLE, L=LQ, K1=.0195; d: DRIFT, L=LD; qd: QUADRUPOLE, L=LQ, K1=-.0195; fodo: line = (qf,d,qd,qd,d,qf); use, period=fodo; match,sequence=FODO; SELECT,FLAG=SECTORMAP,clear; SELECT,FLAG=TWISS,column=name,s,betx,alfx,bety,alfy,mux,muy; TWISS,SAVE; PLOT,interpolate=true,,colour=100,HAXIS=S, VAXIS1=BETX,BETY; PLOT,interpolate=true,,colour=100,HAXIS=S, VAXIS1=ALFX,ALFY; stop; half quad K1=1/(2f) 24.7m vs. 26.4m build FODO cell force periodicity calculate Twiss parameters E. Prebys, NIU Phy 790 Guest Lecture We could calculate α(s),β(s), and γ(s) by hand (slide 25) , but… There have been and continue to be countless accelerator modeling programs; however MAD (“Methodical Accelerator Design”), started in 1990, continues to be the “Lingua Franca”

50. Beam Sizes We have divided out the “π” E. Prebys, NIU Phy 790 Guest Lecture We normally use 95% emittance at Fermilab, and 95% normalized emittance of the beam going into the Main Ring was about 12 π-mm-mr, so the normalized RMS emittance would be We combine this with the equations (slide 45), beam parameters (slide 47) and lattice functions (slide 48) to calculate the beam sizes at injection and extraction.